Number Sense

CAHSEE on Target 

UC Davis, School and University Partnerships 


Mathematics Curriculum 

Published by 

The University of California, Davis, 

School/University Partnerships Program 

2006 

Director 

Sarah R. Martinez, School/University Partnerships, UC Davis 

Developed and Written by 

Syma Solovitch, School/University Partnerships, UC Davis 

Editor 

Nadia Samii, UC Davis Nutrition Graduate 

Reviewers 

Faith Paul, School/University Partnerships, UC Davis 

Linda Whent, School/University Partnerships, UC Davis 

The CAHSEE on Target curriculum was made possible by 

funding and support from the California Academic Partnership Program, 

GEAR UP, and the University of California Office of the President. 

We also gratefully acknowledge the contributions of teachers 

and administrators at Sacramento High School and Woodland High School 

who piloted the CAHSEE on Target curriculum. 

© Copyright The Regents of the University of California, Davis campus, 2005-06 

All Rights Reserved. Pages intended to be reproduced for students activities 

may be duplicated for classroom use. All other text may not be reproduced in any form 

without the express written permission of the copyright holder. 

For further information, 

please visit the School/University Partnerships Web site at: 

http://sup.ucdavis.edu


UC Davis, School/University Partnerships 

Student Workbook: Number Sense Strand 

Introduction to the CAHSEE 

The CAHSEE stands for the California High School Exit Exam. The 

mathematics section of the CAHSEE consists of 80 multiple-choice 

questions that cover 53 standards across 6 strands. These strands 

include the following: 

Number Sense (14 Questions) 

Statistics, Data Analysis & Probability (12 Questions) 

Algebra & Functions (17 Questions) 

Measurement & Geometry (17 Questions) 

Mathematical Reasoning (8 Questions) 

Algebra 1 (12 Questions) 

What is CAHSEE on Target? 

CAHSEE on Target is a tutoring course specifically designed for the 

California High School Exit Exam (CAHSEE). The goal of the program 

is to pinpoint each student’s areas of weakness and to then address 

those weaknesses through classroom and small group instruction, 

concentrated review, computer tutorials and challenging games. 

Each student will receive a separate workbook for each strand and will 

use these workbooks during their tutoring sessions. These workbooks 

will present and explain each concept covered on the CAHSEE, and 

introduce new or alternative approaches to solving math problems. 

What is Number Sense? 

Number Sense is the understanding of numbers and their 

relationships. The Number Sense Strand concepts that are tested on 

the CAHSEE can be divided into five major topics: Integers & 

Fractions; Exponents; Word Problems; Percents; and Interest. These 

topics are presented as separate units in this workbook. 

1




Unit 1: Integers & Fractions 

On the CAHSEE, you will be given several problems involving rational 

numbers (integers, fractions and decimals). 

Integers are whole numbers; they include . . . 

• positive whole numbers {1, 2, 3, . . . } 

• negative whole numbers {−1, −2, −3, . . . } and 

• zero {0}. 

Positive and negative integers can be thought of as opposites of one 

another. 

A. Signs of Integers 

All numbers are signed (except zero). They are either positive or 

negative. 

When adding, subtracting, multiplying and dividing integers, we need 

to pay attention to the sign (+ or -) of each integer. 

Example: 5 ● -3 = ____ 

Example: -5 + 4 = ___ 

Example: -3 ● 12 = ___ 

Whether it’s written or not, every number has a sign: 

Example: 5 means +5 

2




Signed Numbers in Everyday Life 

Signed numbers are used in everyday life to describe various 

situations. Often, they are used to indicate opposites: 

Altitude: The elevator went up 3 floors (+3) and then went down 5 

floors (-5). 

Weight: I lost 20 pounds (-20) but gained 10 back (+10). 

Money: I earned $60 (+60) and spent $25 (-25). 

Temperature: The temperature rose 5 degrees (+5) and then fell 2 

degrees (-2). 

Sea Level: Jericho, the oldest inhabited town in the world, lies 853 

feet below sea level (-853), making it the lowest town on earth. 

Mount Everest is the highest mountain in the world, standing at 

8850 meters (+8850), nearly 5.5 miles above sea level. 

Can you think of any other examples of how signed numbers are used 

in life? 

________________________________________________________ 

________________________________________________________ 

________________________________________________________ 

________________________________________________________ 

________________________________________________________ 

________________________________________________________ 

3




i. Adding Integers 

When adding two or more integers, it is very important to pay 

attention to the sign of each integer. Are we adding a positive or 

negative integer? We can demonstrate this concept with a number 

line. 

Look at the two examples below. In the first example, we add a 

positive 3 (+3) to 2. 

Example: 2 + 3 = __ 

In this second example, we add a negative 3 (-3) to 2. 

Example: 2 + (–3) = __ 

As you can see, we get a very different answer in this second problem 

To add integers using a number line, begin with the first number in 

the equation. Place your finger on that number on the number line. 

Look at the value and sign of the second number: if positive, 

move to the right; if negative, move to the left. (If a number 

does not have a sign, this means it is positive.) With your finger, 

move the number of spaces indicated by the second number. 

Example: 1 + (-2) = ___ 

4

On Your Own 

-2 + (-3) = ___ 

-6 + (3) = ___ 

3 + (-6) = ___ 

-3 + 6 = ___ 




5




Rules for Adding Signed Numbers (without a Number Line) 

A. Same Signs 

• Find the sum 

• Keep the sign 

B. Different Signs 

• Find the difference 

• Keep the sign of the larger number (# with larger absolute 

value) 

On Your Own 

-8 + (-7) = ___ -8 + 7 = ___ 

(-13) + (-9) = ___ (+13) + (+9) = ___ 

21 + (-21) = ___ (-21) + 21 = ___ 

–13 +18 = ___ -18 + 13 = ___ 

Add -10 and -5: ___ Add (-10), (+4), and (-16): ___ 

6




ii. Subtracting Integers 

We can turn any subtraction problem into an addition problem. Just 

change the subtraction sign (-) to an addition sign (+) and 

change the sign of the second number. Then solve as you would 

an addition problem. 

Example: –2 - (+ 3) = ___ 

Turn it from a subtraction problem to an addition problem; then 

change the sign of the second number: 

–2 - (+ 3) = -2 + (-3) 

Now solve as you would an addition problem. 

We can show this on a number line. Place your finger on that number 

on the number line. Look at the value and sign of the second number: 

if positive, move to the right; if negative, move to the left. With 

your finger, move the number of spaces indicated by the second 

number: 

Let's look at another problem: 

Example: -2 - (-3) = -2 + (___) 

Answer: ___ 

Answer: ___ 

7

On Your Own 




6 – (+3) = 6 + (___) = _____ 

3 – (-3) = 3 + (___) = ____ 

-5 – (+1) = -5 + (___) = ____ 

1 – (+1) = 1 + (___) = ___ 

8




Rules for Subtracting Signed Numbers (without a Number Line) 

Add its opposite! Draw the line and change the sign (of the second 

number), and follow the rules for addition. 

Example: 6 – (-4) 

Steps: 

• Draw the line (to turn the minus sign into a plus sign): 6 + ____ 

• Change the sign of the second number: 6 + (+ 4) 

• Now you have an addition problem. Follow the rules of adding 

numbers: 6 + 4 =10 

On Your Own: Draw the line and change the sign. Then solve the 

addition problem. 

19 – (- 13) = ___ -17 – (-15) = ___ 

34 – (-9) = ___ -18 - 14 = ___ 

-15 - (-35) = ___ 13 - (+15) = ___ 

-13 – 15 = ___ -35 - (+35) = ___ 

Subtract (-15) from (20): ___ Subtract 4 from (-14): ___ 

9




Signed Numbers Continued 

Look at the following problem: 

Example: 1 - 3 + 5 = ___ 

We can represent this problem on a number line: 

We begin at 1, move 3 spaces backwards (to the left) and then 5 

spaces forwards (to the right). We arrive at + 3. 

When we are given a problem with three or more signed integers, we 

must work out, separately, the addition and subtraction for each 

integer pair: 

1 - 3 + 5 = 1 - 3 + 5 

1 - 3 = -2 ← ⎯⎯ Work out the addition or subtraction for the 1st 2 integers 

-2 + 5 = ___ ← ⎯⎯ Take the answer from above & add it to the last integer. 

On Your Own 

1. 12 + 3 - 5 + 4 = _____ 

2. -3 + 5 - 2 + 3 = _____ 

3. 4 - 6 + 3 - 2 = _____ 

10




iii. Multiplying and Dividing with Signed Numbers 

Multiplying 

The product of two numbers with the same sign is positive. 

Example: -5 ● -3 = 15 

The product of two numbers with different signs is negative. 

Example: -5 ● 3 = -15 

Dividing 

The quotient of two numbers with the same sign is positive. 

Example: -15 ÷ -3 = 5 

The quotient of two numbers with different signs is negative. 

Example: -15 ÷ 3 = -5 

On Your Own 

(+8) (–4) = ___ (–7) (7) = ___ 

(–8) (-8) = ___ (+7)(+8) = ___ 

– 36 (-3) = ___ –36 3 = ___ 

11

B. Absolute Value 




The absolute value of a number is its distance from 0. This distance 

is always expressed as a positive number, regardless if the number 

is positive or negative. 

It is easier to understand this by examining a number line: 

The absolute value of 5, expressed as |5|, is 5 because it is 5 units 

from 0. We can see this on the number line above. The absolute value 

of -5, expressed as |-5|, is also 5 because it is 5 units from 0. Again, 

look at the number line and count the number of units from 0. 

On Your Own: Complete the chart. How far from zero is the number? 

Number Absolute Value 

|-16| 

|-115| 

|342| 

|x| 

|-x| 

|-100| 

12




Finding the Absolute Value of an Expression 

On the CAHSEE, you may need to find the absolute value of an 

expression. To do this, . . . 

• Evaluate the expression within the absolute value bars. 

• Take the absolute value of that result. 

• Perform any additional operations outside the absolute value 

bars. 

Example: 3 + |-4 - 3| = 3 + |-7| = 3 + 7 = 10 

On Your Own: Complete the chart. 

5 ●|3-8| = 5 ●|-5| = 5 ● 5 = 25 

|15 + 6| = = 

|-6 + 2| = = 

|1 - 3 + 2| = = 

4 + |-6| = = 

|-4| + |4| = = 

|16| - |-16| = = 

|-2| - |13| = = 

|-2| - |-13| = = 

13




Absolute Value Continued 2.5 

While the absolute value of a number or expression will always be 

positive, the number between the absolute value bars can be 

positive or negative. 

Notice that in each case, the expression is equal to +8. 

You may be asked to identify these two possible values on the 

CAHSEE. 

Example: If |x| = 8, what is the value of x? 

For these types of problems, the answer consists of two values: the 

positive and negative value of the number. 

In the example above, the two values for x are 8 or -8. 

On Your Own 

1. If |y| = 225, what is the value of y? ____ or ____ 

2. If |x| = 1,233, what is the value of x? ______ or ______ 

3. If |m| = 18, what is the value of m? ____ or ____ 

4. If |x| = 12, what is the value of x? ____ or ____ 

5. If |y| = 17, what is the value of y? ____ or ____ 

14

C. Fractions 




A fraction means a part of a whole. 

Example: In the picture below, one of four equal parts is shaded: 

1 

We can represent this as a fraction: 

4 

1 

Fractions are expressed as one number over another number: 

4 

Every fraction consists of a numerator (the top number) and a 

denominator (the bottom number): 

A ← ⎯⎯ Numerator 

B ← ⎯⎯ Denominator 

A 

Fractions mean division: = A ÷ B 

B 

1 

= 1 ÷ 4 = 4 1 = .25 

4 

4 

= 4 ÷ 5 = 5 4 = .8 

5 

1 

= 1 ÷ 2 = 2 1 = .5 

2 

15




i. Adding & Subtracting Fractions 

Same Denominator: Keep the denominator; add the numerators: 

1 2 

Example: + = _____ 

4 4 

We can represent this problem with a picture: Begin with the first 

1 2 

fraction, , and add two more fourths ( ): 

4 

4 

3 

We now have three-fourths of the whole shaded: 

4 

On Your Own: Add the following fractions. 

1 2 

+ = ------- 

8 8 

1 1 

+ = ------- 

3 3 

2 3 

+ = ------- 

5 5 

2 1 

- = ------- 

3 3 

Rule: When adding and subtracting fractions that have common 

denominators, we just add or subtract the numerators and keep the 

denominator. It gets trickier when the denominators are not the same. 

16




Different Denominator 

1 3 

Example: + 

4 8 

Let's represent this with a picture: 

The first picture shows one whole divided into four parts. One 

1 

of these parts is shaded. We represent this as a fraction: 

4 

The second picture shows one whole divided into eight parts. Three 

3 

of these parts are shaded. We represent this as a fraction: 

8 

In order to add these two fractions, we need to first divide them up 

into equal parts. The first picture is divided into fourths but the 

second is divided into eighths. We can easily convert the first 

picture into eighths by drawing two more lines (i.e. divide each fourth 

by half): 

17




Now let's see how the first fraction would appear once it is divided 

into eighths: 

1 2 

We can see, from the above picture, that is equal to . 

4 

8 

Now that we have a common denominator (8), we can add the 

2 3 

fractions: + . Just keep the denominator and add the 

8 8 

numerators: 

2 3 

+ = 

8 8 8 

2 4 

Let's look at another example: + 

3 5 

Can we add these two fractions in their current form? Explain. 

To add two fractions, we need a common denominator. We must 

therefore convert the fractions to ones whose denominator is the 

same. We can use any common denominator, but it is much easier to 

use the lowest common denominator, or LCD. One way to find the 

LCD is to make a table and list, in order, the multiples of each 

denominator. (Multiple means Multiply!) 

18




Finding the Lowest Common Denominator (LCD) 

Look at the last problem again: 2 + 4 

3 5 

Now list the multiples of each denominator until you reach a 

common number. 

Multiples of 3 Multiples of 5 

3 5 

6 10 

9 15 

12 

15 

The lowest common denominator (LCD) is the first 

common number in both columns: 15. This will be the new 

denominator for both fractions. 

Since we changed the denominators, we must also change the 

numerators so that each new fraction is equivalent (or equal) to the 

original fraction. 

• Let’s start with the first original fraction: 2/3. Go back to the 

table. How many times did we multiply the denominator, 3, 

by itself? (Hint: How many rows did we go down in the first 

column?) ___ 

• Since we multiplied the denominator (3) by __ to get 15, 

we must also multiply the numerator (2) by ___. 

Our new fraction is 15 

• Now let’s look at the second fraction: 4/*5. Since we 

multiplied the denominator (5) by ___, we do the same to the 

numerator: 4 ● ___ = ___. 

Our new fraction is 15 

• Now add the new fractions. 15 + 15 = 15 

We have an improper fraction because the numerator > the 

denominator. We must change it to a mixed number: 

22 

= ________ 

15 

19




Let's look at another example: 

Example: Add the following fractions: 3 + 4 

4 5 

In order to add these fractions we must first find a common 

denominator. Make a table and list all of the multiples for each 

denominator until we reach a common multiple: 

Multiples of 4 Multiples of 5 

4 5 

8 10 

12 15 

16 20 

20 

We have a common denominator for both fractions: 20. Since we 

changed the denominators for both fractions, we must also change the 

numerators so that each new fraction is equivalent to the original 

fraction. 

3 

Let’s begin with the first fraction: = 

4 20 

4 

Now let’s proceed to the second fraction: = 

5 20 

Now both fractions have common denominators; add 

them: 

20 + 20 = 20 

If the sum is an improper fraction (i.e. numerator > denominator), 

we generally change it to a proper fraction: _______ 

20

On Your Own 

Example: 3 + 5 

4 6 




Step 1: Make a table and list the multiples of each denominator 

until you reach a common denominator: 

Step 2: Convert each fraction to an equivalent fraction: 

Step 3: Add the fractions: 

Note: If you end up with an improper fraction, be sure to 

convert it to a mixed number. 

21

Practice 

4 2 

+ = ----------- 

5 5 

7 3 

– = ----------- 

9 9 

2 3 

+ = ----------- 

5 4 

2 5 

+ = ----------- 

3 8 

3 1 

- = ----------- 

4 6 

5 1 

- = ----------- 

8 2 




' 

22




Prime Factorization 

Another way to find the lowest common denominator of two fractions 

is through prime factorization. First, let’s learn more about prime 

numbers: 

Prime Numbers: A prime number has two distinct whole number 

factors: 1 and itself. 

Note: 1 is not prime because it does not have two distinct 

factors. 

Example: 6 is not prime because it can be expressed as 2 ● 3. 

Example: 7 is prime because it can be expressed only as the 

product of two distinct factors: 1 ● 7. 

Write the first 10 prime numbers: 

2 3 5 ___ ___ ___ ___ ___ ___ ___ 

Composite Numbers 

A non-prime number is called a composite number. Composite 

numbers can be broken down into products of prime numbers: 

Example: 4 = 2 X 2 

Example: 12 = 2 X 6 = 2 X 2 X 3 

Example: 66 = 6 X 11 = 2 X 3 X 11 

Example: 24 = 2 X 12 = 2 X 2 X 2 X 3 

Example: 33 = 3 X 11 

Example: 125 = 5 X 5 X 5 

23




Practice: Circle all of the prime numbers in the chart below: 

1 

6 

11 

16 

21 

26 

31 

36 

41 

46 

2 

7 

12 

17 

22 

27 

32 

37 

42 

47 

3 

8 

13 

18 

23 

28 

33 

38 

43 

48 

4 

9 

14 

19 

24 

29 

34 

39 

44 

49 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

24

Prime Factor Trees 




We can find the prime factors of a number by making a factor 

tree: 

Example: Find the prime factors of 18. 

• Write your number: 18 

• Begin with the smallest prime number factor of 18 (i.e. the 

smallest prime number that divides evenly 18. 

This number is 2. 

• Draw two branches: 2 and the second factor: 9. 

18 

⁄ \ 

2 9 

• Continue this process for each branch until you have no remaining 

composite numbers. The prime factors of 18 are the prime 

numbers at the ends of all the branches: 

18 

⁄ \ 

2 9 

⁄\ 

3 3 

The prime factored form of 18 is ___ ● ___ ● ___. 

25




Example: Find the prime factors of 60 using the factor tree: 

60 

⁄\ 

2 30 

⁄\ 

2 15 

⁄\ 

3 5 

The prime factors of 60 are the factors at the end of each 

branch: ___, ___, ____ and ___. 

Helpful Guidelines: 

• Start with the smallest numbers: first 2’s, then 3’s, and so on. 

• If a number is even, it is divisible by 2. 

Note: An even number ends in 0, 2, 4, 6, and 8. 

Examples: 124 38 46 180 112 

• If the digits of a number add up to a number divisible by 3, the 

number is divisible by 3. 

Example: 123 can be divided evenly by 3 because if we add all of its 

digits, we get 6: 1 + 2 + 3 = 6 

Since the sum of the digits of 123 is divisible by 3, so too is 123. 

• If a number ends in 0 or 5, it is divisible by 5. 

Examples: 25 130 125 455 

26

On Your Own 




Find the prime factors of each number, using a factor tree: 

64 48 

72 

27




Prime Factorization and the Lowest Common Denominator 

On the CAHSEE, you will be asked to find the prime factored form of 

the lowest common denominator (LCD) of two fractions: 

Example: Find the prime factored form for the lowest 

5 5 

common denominator + . 

6 9 

There are two methods we can use to solve this problem: 

Method I: Factor Tree and Pairing 

Steps: 

• Make a factor tree for both denominators: 

6 9 

⁄\ ⁄\ 

2 3 3 3 

• Pair up common prime factors: 

• Multiply the common factor (counted once) by all leftover 

(unpaired) factors: 

LCD = 3 ● __ ● ___ = ____ 

28




Let's look at another example: 

Example: Find the least common multiple of 72 and 24. Write the 

LCM in prime-factored form. 

Steps: 

• Make a factor tree for each number: 

72 24 

⁄\ ⁄ \ 

2 36 2 12 

⁄\ ⁄\ 

2 18 2 6 

⁄ \ ⁄\ 

2 9 2 3 

⁄ \ 

3 3 

• Pair off common factors: 

72 = 2 ● 2 ● 2 ● 3 ● 3 

24 = 2 ● 2 ● 2 ● 3 

←⎯⎯ Count any common factor once! 

• Multiply all common factors by all leftover (unpaired) factors: 

LCM = __ ● __ ● __ ● __ ● ___ = ____ 

29




On Your Own: Solve the following problems, using the factor 

tree/pairing method. 

1. What is the prime factored form of the lowest common 

5 5 

denominator of + ? 

9 12 

2. Find the least common multiple, in prime-factorization form, 

of 12 and 15. 

We will now look at the second method to find the prime factored 

from of the lowest common denominator (LCD) of two fractions. 

30




Method II: Factor Tree and Venn Diagram 

To illustrate this second method, let's return to the original problem: 

Example: Find the prime factored form for the lowest 

5 5 

common denominator of + . 

6 9 

• Use the factor tree method to find the prime factored form of 6: 

6 

⁄ \ 

2 3 

• Use the factor tree method to find the prime factored from of 9: 

9 

⁄ \ 

3 3 

• Use a Venn diagram to find the prime-factored form of the lowest 

common denominator: 

On the next page, we will learn how to fill out this diagram. 

31

Venn Diagrams 




Venn diagrams are overlapping circles that help us compare and 

contrast the characteristics of different things. We can use them to 

find what is common to two items (where the circles overlap in the 

middle) and what is different between them (what is outside the 

overlap on either or both sides). 

Here, we want to find out which prime factors are the same for two 

numbers and which factors are distinct, or different. 

6 9 

⁄ \ ⁄ \ 

2 3 3 3 

Steps: 

• Since only one 3 is common to both numbers, we need to put it in 

the middle, where the two circles overlap: 

6 Both 9 

↓ 

Continued on next page ⎯ 

⎯→ 

32




• Now find the prime factors that are left for 6 and place them in the 

part of the circle for 6 that does not overlap with the circle for 9. 

6 Both 9 

↓ 

• Next, find the prime factors that are left for 9 and place them in the 

part of the circle that does not overlap with the circle for 6. 

6 Both 9 

↓ ↓ ↓ 

• The lowest common denominator for 6 and 9 is the product of all 

of the numbers in the circles: 

___ ● ___ ● ___, which is equal to ____ 

Note: To write the LCD in prime-factored form, we do not carry out 

the multiplication; we just write the prime numbers: 

LDC of 6 and 9 = ___ ● ___ ● ___ 

33




On Your Own 

1. What is the prime factored form of the lowest common denominator 

1 3 

of and ? 

6 10 

• Create separate prime factor trees for both denominators: 

6 10 

⁄ \ ⁄ \ 

__ __ __ __ 

• Organize the prime factors of both denominators, using a Venn 

diagram: 

6 Both 10 

What is the LCD? ________ 

Write the LCD in prime factored form: _____________ 

34




2. Find the prime factored form of the lowest common denominator for 

the following: 

5 11 

+ 

8 12 

Factor Trees: 

LCD: ______ 

8 Both 12 

LCD in prime factored form: _________________ 

35




ii. Multiplying Fractions 

Whenever you are asked to find a fraction of a number, you need to 

multiply. In math, the word “of” means multiply. 

1 1 

Example: Find of . 

2 2 

1 1 

This is a multiplication problem. It means, “What is ● ?” 

2 2 

We can represent the problem visually. Here is the first part of the 

1 

problem: of the circle has been shaded. 

2 

1 

Taking of a number means dividing it by 2. 

2 

Now, if we take one-half of this again (divide it by 2 again), we get 

the following: 

1 1 1 

of is equal to . 

2 2 

4 

We end up with one-fourth of the circle. 

Note: We also could have solved the above problem by multiplying 

the numerator by the numerator and the denominator by the 

denominator: 

Numerator ● Numerator_ = 1 ● 1 = 1 

Denominator ● Denominator 2 ● 2 4 

36




When working these problems out during the CAHSEE, you will need to 

apply this rule: 

Numerator ● Numerator__ 

Denominator ● Denominator 

Look at the next problem: 

1 1 

Find of 24. In math, we can write this as follows: ● 24 

2 

2 

The first factor is a fraction and the second factor is a whole number. 

We can easily change the second factor to a fraction because any 

whole number can be expressed as a fraction by placing it over a 1: 

24 

24 = because 24 means 24 ones. 

1 

1 24 

We can rewrite the problem as follows: ● 

2 1 

Now, just follow the rule for multiplying two fractions: 

Numerator ● Numerator__ 

Denominator ● Denominator 

1 24 24 

● = = ___ 

2 1 2 

1 

Note: Taking of 24 means dividing 24 by 2. 

2 

37




Now look at the next example: 

24 5 

Example: ● = ____ 

1 6 

There are two ways to solve this problem: 

1. The hard way: Perform all operations 

• Multiply numerators: 24 ● 5 

• Multiply denominators: 1 ● 6 

• Divide new numerator by denominator: 120 ÷6 

24 ● 5 = 120 = 120 ÷6 = ___ 

1 ● 6 6 

2. The easy way: Simplify first, and then multiply: 

4 

24 ● 5_ = ____ Simplify by dividing out common factors! 

1 6 1 

Look at the following problems: 

533 9 3 

● 4 = ______ 3,435 ● = _____ 79 ● = _____ 

4 

9 

3 

Do you need to work out these problems, or do you already know the 

answers? ________________________________________________ 

________________________________________________________ 

Remember: If you divide both a numerator and denominator by a 

common factor, you can make the problem much simpler to solve. So 

save yourself the time and work, and recognize these types of 

problems right away. 

38




Look at the next set of problems: 

4 6 

● 

12 8 

8 5 14 3 

● ● 

15 12 

21 7 

What do you notice about the above problems? __________________ 

________________________________________________________ 

There is a lot of heavy multiplication involved in these problems. Is 

there a way to make your work easier? Explain: 

We can _________ fractions by ____________________ 

before solving. 

We can simplify these problems quite a bit before solving. This makes 

our job easier. Let’s look at the first problem: 

4 6 

● 

12 8 

We can divide out common factors in each fraction. These 

common factors become clear if we write each fraction as a 

product of prime factors. Let's begin with the first fraction: 

4 = 1 2 ● 1 2____ = 1 

12 12 ● 12 ● 3 3 

Now do the second fraction on your own: 

6 

= ____________________ 

8 

Now let's multiply the two reduced fractions; but first, can we simplify 

anymore? ______ If so, simplify first, and then multiply: 

39




On Your Own: Simplify and solve: 

8 5 

● = ______ 

15 12 

3 10 

● = ______ 

5 21 

14 3 

● = ______ 

21 7 

12 5 

● = ______ 

15 6 

2 2 

36 ● = _____ 27 ● = _____ 

3 

9 

2 3 

● = _____ 

3 2 

10 3 

● = _____ 

15 2 

40




iii. Dividing Fractions 

When you divide something by a fraction, think, “How many 

times does the fraction go into the dividend?” 

1 

Example: 3 ÷ 

2 

↑ 

dividend 

This means, “How many times does ___ go into ____?” 

We can represent this visually: 

Answer: __________ 

1 

Example: 2 ÷ 

8 

We can represent this visually: 

Answer: __________ 

41




On Your Own: Solve the next few problems, asking each time, 

“How many times does the fraction go into the whole number?” 

1 

3 ÷ = ___ 

4 

1 

3 ÷ = ___ 

8 

1 

4 ÷ = ___ 

8 

Do you see a pattern? Explain. 

___________________________________________________ 

___________________________________________________ 

42

Reciprocals 




As we saw in the previous exercise, each time we divide a whole 

number by a fraction, we get as our answer the product of the 

whole number and the reciprocal of the fraction. 

Reciprocal means the flip-side, or inverse. 

4 5 

Example: The reciprocal of is . 

5 4 

On Your Own: Find the reciprocal of each fraction: 

3 7 12 

⎯ ⎯→ _____ ⎯ ⎯→ _____ ⎯ ⎯→ _____ 

4 

9 

5 

13 35 1 

⎯ ⎯→ ____ ⎯ ⎯→ ____ ⎯ ⎯→ _____ 

1 

53 

12 

Now let's find the reciprocal of a whole number. We know that 

any whole number (or integer) can be expressed as a fraction by 

placing it over 1: 

35 

Example: 35 = 

1 

The reciprocal is the fraction turned upside down, or inverted: 

1 

Example: The reciprocal of 35 is 

35 

On Your Own: Find the reciprocal of each integer. 

1000 121 173 -18 -100 

_____ ____ ____ ____ ____ 

43




Now we are ready to divide a whole number by a fraction. 

1 2 10 20 

Example: 2 ÷ = ● = = 20 

10 1 1 1 

We can represent the above problem visually: 

1 

2 ÷ means . . . 

10 

If we count the number of little rectangles in the two big 

rectangles, we get _____. 

On Your Own 

1 

3 ÷ = ________________________ 

5 

1 

6 ÷ ÷ = ________________________ 

5 

1 

5 ÷ = _________________________ 

3 

1 

2 ÷ = _________________________ 

3 

1 

4 ÷ = _________________________ 

2 

44




Simplifying Division Problems 

3 9 

Example: ÷ 

5 10 

Remember the rule for dividing fractions: 

Rule: When dividing fractions, multiply the first fraction 

by the reciprocal of the second fraction! 

Steps: 

• Multiplying the first fraction by the reciprocal of the second 

fraction, we get . . . 

3 10 

● 

5 9 

• We can simplify this problem by dividing out common factors: 

1 3 ● 10 2 

1 5 9 3 

• Now, apply the rule for multiplication: 

Numerator ● Numerator____ = 1 ● 2 = ____ 

Denominator ● Denominator 1 ● 3 

45




On Your Own: Simplify and solve. 

1 3 ÷ = _________________ 

4 8 

2 3 ÷ = _________________ 

5 10 

3 9 ÷ = _________________________ 

7 14 

5 15 ÷ = __________________ 

8 24 

1 13 ÷ = __________________ 

10 10 

46




Unit Quiz: The following problems appeared on the CAHSEE. 

11 1 1 

1. – ( + ) = 

12 3 4 

1 

A. 

3 

3 

B. 

4 

5 

C. 

6 

9 

D. 

5 

5 7 

2. Which fraction is equivalent to + ? 

6 8 

35 

A. 

48 

6 

B. 

7 

20 

C. 

21 

41 

D. 

24 

3. What is the prime factored form for the lowest common 

2 7 

denominator of the following: + ? 

9 12 

A. 3 X 2 X 2 

B. 3 X 3 X 2 X 2 

C. 3 X 3 X 3 X 2 X 2 

D. 9 X 12 

47




4. Which of the following is the prime factored form of the lowest 

7 8 

common denominator of +15 ? 

10 

A. 5 X 1 

B. 2 X 3 X 5 

C. 2 X 5 X 3 X 5 

D. 10 X 15 

5. Which of the following numerical expressions results in a negative 

number? 

A. (-7) + (-3) 

B. (-3) + (7) 

C. (3) + (7) 

D. (3) + (-7) + (11) 

6. One hundred is multiplied by a number between 0 and 1. The 

answer has to be ____. 

A. less than 0. 

B. between 0 and 50 but not 25. 

C. between 0 and 100 but not 50. 

D. between 0 and 100. 

7. If |x| = 3, what is the value of x? 

A. -3 or 0 

B. -3 or 3 

C. 0 or 3 

D. -9 or 9 

48




8. What is the absolute value of -4? 

A. -4 

1 

B. − 

4 

1 

C. 

4 

D. 4 

9. The winning number in a contest was less than 50. It was a 

multiple of 3, 5, and 6. What was the number? 

A. 14 

B. 15 

C. 30 

D. It cannot be determined 

10. If n is any odd number, which of the following is true about n + 1? 

A. It is an odd number. 

B. It is an even number 

C. It is a prime number 

D. It is the same as n −1. 

11. Which is the best estimate of 326 X 279? 

A. 900 

B. 9,000 

C. 90,000 

D. 900,000 

49




12. The table below shows the number of visitors to a natural history 

museum during a 4-day period. 

Day Number of Visitors 

Friday 597 

Saturday 1115 

Sunday 1346 

Monday 365 

Which expression would give the BEST estimate of the total number 

of visitors during this period? 

A. 500 + 1100 + 1300 + 300 

B. 600 + 1100 + 1300 + 300 

C. 600 + 1100 + 1300 + 400 

D. 600 + 1100 + 1400 + 400 

2 

13. John uses of a cup of oats per serving to make oatmeal. How 

3 

many cups of oats does he need to make 6 servings? 

2 

A 2 

3 

B 4 

1 

C 5 

3 

D 9 

14. If a is a positive number and b is a negative number, which 

expression is always positive? 

A. a - b 

B. a + b 

C. a X b 

D. a ÷ b 

50

Unit 2: Exponents 




On the CAHSEE, you will be given several problems on exponents. 

Exponents are a shorthand way of representing how many times a 

number is multiplied by itself. 

Example: 9 9 9 9 can be expressed as 9 4 since four 9's are 

multiplied together. 

Base ←⎯⎯ 9 4 exponent 

The number being multiplied is called the base. 

The exponent tells how many times the base is multiplied by itself. 

9 4 is read as “9 to the 4 th power,” or “9 to the power of 4.” 

Let's look at another example: 2 = 2 ● 2 ● 2 ● 2 ● 2 = 32 

On Your Own 

2³ = ___ 2 = ___ 

3² = ___ 3³ = ___ 

Power of 0 

Any number raised to the 0 power (except 0) is always equal to 1. 

Example: 100 0 = 1 

On Your Own 

7 0 = ___ 293 0 = ___ (-131) 0 = ___ 47 0 = ___ 

51

Power of 1 




A number raised to the 1 st power (i.e., an exponent of 1) is always 

equal to that number. 

Example: 100 1 = 100 

On Your Own 

7 1 = ____ 293 1 = ____ (-131) 1 = ____ 47 1 = ____ 

Power of 2 (Squares) 

A number raised to the 2 nd power is referred to as the square of a 

number. When we square a whole number, we multiply it by itself. 

Example: 12² = 12 ● 12 = 144 

The square of any whole number is called a perfect square. 

Here are the first 3 perfect squares: 

1² = 1 ● 1 = 1 2² = 2 ● 2 = 4 3² = 3 ● 3 = 9 

On Your Own: Write the perfect squares for the following numbers: 

4² = ____ 5² = ____ 6² = ____ 7² = ____ 

8² = ____ 9² = ____ 10² = ____ 11² = ____ 

20² = ____ (2 - 8)² – (3 - 7)² =_______ 3² + 5² = _____ 

52

Square Roots 




The square root ( ) of a number is one of its two equal factors. 

Example: 8² = 64 

1 2 3 4 5 6 7 8 

2 

3 

4 

5 

6 

7 

8 

Any number raised to the second power (the power of 2) can be 

represented as a square. That’s why it’s called “squaring the 

number.” 

The square above has 64 units. Each side (the length and width) is 8 

units. The area of the square is determined by multiplying the length 

(8 units) by the width (8 units). The square root is the number of 

units in each of the two equal sides: 8 

Note: 64 has a second square root: -8 (-8 ● -8 = +64). However, 

when we are asked to evaluate an expression, we always take the 

positive root. 

Example: Find the square root of 36. 

Answer: 36 = ___ 

53

On Your Own 

1. 25 = ___ 

2. 16 = ___ 

3. 100 = ___ 

4. 81 = ___ 

5. 49 = ___ 

6. 121 = ___ 

7. 400 = ___ 

8. 4 + 9 = ___ 

9. 




2 2 

3 + 4 = ___ 

10. Which is not a perfect square? 

A. 144 

B. 100 

C. 48 

D. 169 

54




Power of 3 (Cubes) 

A number with an exponent of 3 (or a number raised to the 3 rd power) 

is the cube of a number. 

Example: 5³ = 5 ● 5 ● 5 = 125 

The cube of a whole number is called a perfect cube. 

Cubes of Positive Numbers 

The cube of a positive number will always be a positive number. 

1³ = 1 ● 1 ● 1 = 1 2³ = 2 ● 2 ● 2 = 8 

Cubes and Negative Numbers 

The cube of a negative number will always be a negative number. 

(-1)³ = (-1)(-1)(-1) = -1 (-2) 3 = (-2)(-2)(-2) = -8 

On Your Own: Write the perfect cubes for the following numbers: 

3³ = _____ 4³ = ______ 5³ = _____ 

-3³ = _____ -4³ = ______ -5³ = _____ 

55

Cube Roots 




The cube root of a number is one of its three equal factors. The cube 

root of a positive number will always be a positive number. 

Example: What is the cube root of 27? 

The cube root of 27 is written as 3 27 

To find the cube root of 27, ask, “What number multiplied by itself 3 

times is equal to 27?” 

__ ● __ ● __ = 27 

3 ● 3 ● 3 = 27, or 3³ = 27. 

On Your Own 

3 3 3 3 

8 = ___ 64 = ____ 1 , 000 = ____ 125 = ____ 

Cube Roots of Negative Numbers 

The cube root of a negative number will always be a negative number. 

Example: 3 − 64 

Ask, “What number multiplied by itself 3 times is equal to -64? 

__ ● __ ● __ = -64 

-4 ● -4 ● -4 = -64, or (-4)³ = -64 

On Your Own: 3 − 1, 

000 = ____ 3 − 125 = ____ 3 − 8 = ____ 

56




Raising Fractions to a Power 

When raising a fraction to an exponent, both the numerator (top 

number) and the denominator (bottom number) are raised to the 

exponent: 

2 

5 

numerator 

denominator 

Example: (2/5)³ = 2 ● 2 ● 2 = 8_ 

5 ● 5 ● 5 125 

On Your Own 

(1/4)²= ____ (1/2) = _____ (2/3)³ = ____ 

(2/5)³ = _____ (4/7)² = ____ (6/11)² = _____ 

(3/4)³ = ____ (5/12)= ____ (2/5)² = ____ 

Taking the Root of a Fraction 

When taking the root of a fraction, we must take the root of both the 

numerator and denominator. 

Example: 

On Your Own 

3 

9 25 36 

= _____ = _____ = _____ 

16 

100 

81 

1 81 1 

= ____ = _____ = _____ 

125 

121 

9 

3 

3 

27 

= _____ 

64 

8 

= _____ 

27 

57




Raising Negative Numbers to a Power 

When raising a negative number to a power, we are raising both the 

number and the negative sign. The answer may be positive or 

negative: 

A. If the exponent is an even number, the answer will be a positive 

number (as in the example below) since a negative multiplied by a 

negative equals a positive. 

Example: (-2) 6 = (-2)(-2)(-2)(-2)(-2)(-2) = 64 

Note: (-2) 6 and -2 6 are two different problems: 

• -2 6 tells us to multiply positive 2 by itself 6 times 

(2)(2)(2)(2)(2)(2) = 64 

and then take the negative of that answer: -64 

• (-2) 6 tells us to multiply -2 by itself 6 times: 

(-2)(-2)(-2)(-2)(-2)(-2) = +64 

B. If the exponent is an odd number, the answer will be a negative 

number (a positive multiplied by a negative equals a negative). 

Example: (-2) 5 = (-2)(-2)(-2)(-2)(-2) = -32 

On Your Own 

(-3)² = _____ -3³ = _____ (-2)² = _____ (-2) = _____ 

-4² = _____ -4 = _____ (-5)² = _____ -5² = _____ 

58




Negative Exponents 

On the CAHSEE, you may be given an expression with a negative 

exponent. When an exponent is negative, the expression represents 

a fraction: 

1 

Example: 3¯³ means 3 

3 

Notice that the exponent is now positive. 

Remember: Any whole number can be written as a fraction by placing 

it over 1: 

3 

3¯³ can also be written as 

1 

3 − 

We now flip the fraction and make the exponent positive. 

3 3 − 

1 

1 

⎯ ⎯→ 3 

3 

In the above example, the numerator is equal to 1, and the 

denominator consists of the base and the (positive) exponent. 

On Your Own: Flip the fraction and make the exponent positive. 

3¯² = ____ 5¯³= ____ 2¯³ = ____ 4¯² = ____ 

1 

59




Negative Exponents and Fractions 

When raising a fraction to a negative exponent, just invert the entire 

fraction and make the exponents positive: 

Example: − 3 

3 

1 

Here we have a fraction whose denominator consists of the base and a 

negative exponent. If we invert the fraction, the exponent 

becomes positive: 

3 

1 

−3 

3 

⎯ ⎯→ 

1 

3 

On Your Own 

1 

1. ( ) 3 

2 

2. ( ) 3 

1 

3. ( ) 8 

3 

4. ( ) 5 

1 

5. ( ) 3 

−2 

−1 

−2 

−2 

−3 

⎯ ⎯→ 3 3 

2 

= ( ) 

= ___ 

1 

= ( ) = ___ 

= _____ 

= _____ 

= _____ 

60




Multiplying Expressions Involving Exponents with a Common 

Base 

On the CAHSEE, you may be asked to multiply expressions involving 

exponents. 

Example: 3 5 ● 3 4 

In order to multiply expressions involving exponents, there must be a 

common base. In the above example, the base (3) is common to both 

terms. 

When we have a common base, the rule for multiplying the 

expressions is simple: keep the base and add the exponents: 

Base ● Base = Base 5+4 ⎯ ⎯→ 3 5 ● 3 4 =3 5+4 =3 9 

On Your Own 

2² ● 2⁸ = ____ 3¹ ● 3⁷ = ____ 

4 3 ● 4 2 = ____ 4 ● 4¯ = ____ 

3¯³ ● 3³ = ____ 6 ● 6 = ____ 

61




Note: In some cases, we may end up with a negative exponent. 

Remember to apply the rules for negative exponents: invert the 

fraction and make the exponent positive. 

Example: 3 2 ● 3 -3 = 3 2+(-3) 3 

= 3¯ = 

1 

1 − 

On Your Own 

1 1 

= = 1 

3 3 

4 1 ● 4¯ 3 = _______ 5¯ 1 ● 5¯ 1 = __________ 

3 5 ● 3¯ 8 = _______ 4 ● 4¯ 7 = __________ 

Dividing Expressions Involving Exponents with a Common Base 

On the CAHSEE, you may be asked to divide expressions involving 

exponents. 

For these types of problems, there must be a common base: 

3 

5 

Example: 3 

3 

To divide exponents with a common base, keep the base and 

subtract the exponent in the denominator from the exponent in the 

numerator: 

5 

5 

Base - 

= Base = Base² ⎯ ⎯→ 

3 

3 

Base 

3 5-3 2 

= 3 = 3 

3 

62

On Your Own 




3 

5 

3 

= _______ 2 

3 

3 

4 

8 

5 

= _________ 2 

3 

5 

4 

= ________ 

4 

2 

= ________ 

2 

−2 

3 

The next problem is a bit more complicated: −3 

3 

Remember, when an exponent is negative, the expression is a 

fraction and the numerator (top number) is always equal to 1, 

while the denominator (bottom number) is the base. But, here, the 

problem is already a fraction, so we really have one fraction over 

another fraction. We will get back to the above problem in a moment, 

but first, let’s do a quick review of the rules for dividing fractions: 

Dividing Fractions 

To divide two fractions, we multiply the 1st fraction by the 

reciprocal of the 2 nd fraction. This means that we invert the second 

fraction over, or invert it. 

2 5 

For example, the reciprocal of is . 

5 2 

Let’s solve the following problem: 

3 1 3 3 9 

÷ = ● = 

4 3 4 1 4 

9 

is an improper fraction (numerator > denominator). We must 

4 

1 

change this to a mixed fraction (whole number and fraction): 2 

4 

63




−2 

3 

Now we are ready to tackle the earlier: problem: −3 

3 

We have negative exponents in both the numerator and the 

denominator. We can therefore rewrite each as fractions with 

positive exponents: 

1 1 

3¯² = and 3¯³ = 2 

3 

3 

3 

We can now rewrite the original problem as follows: 2 

3 

1 1 

÷ 3 

Applying the rule for dividing two fractions, we invert the second 

fraction and multiply: 

1 3 

● 2 

3 1 

3 

Multiplying the numerator by the numerator, and the denominator by 

3 

3 

the denominator, we get . . . 2 

3 

Now we apply the rules for dividing exponents: the base remains the 

same and we subtract the exponent in the denominator from the 

exponent in the numerator: 

3 

3 

= 3³¯² = ________ 

2 

3 

Shortcut! 

Since both exponents in the above example are negative, a quicker 

way to solve the problem is to just flip the fractions and reverse the 

sign of each exponent; then simplify: 

3 

− 

3 

−2 

3 

3 

3 

= = _________ 

2 

3 

3 

64

Another Shortcut! 




A third way to solve the problem is to apply the rule for dividing 

expressions with exponents: keep the base and subtract the 

exponent in the denominator from the exponent in the numerator: 

−2 

Base -2 -(-3) - + 3 

= Base = Base ² ⎯→ 

−3 

Base 

3 

− 

3 

−2 

3 

On Your Own 

2 

− 

2 

−8 

8 

− 

8 

3 

−4 

2 

− 

2 

2 

−7 

2 

− 

2 

2 − 

2 

1 

2 

3 

2 

3 

⎯ Base 1 

= 3 -2 -(-3) = 3 - ² + 3 ⎯ ⎯→ 3 1 = 3 

= ___________ 

= ___________ 

= ___________ 

= ________ 

= ___________ 

65




Power Raised to a Power 

When raising a power to a power, multiply the powers together: 

Example: (2 3 ) 2 = 2 (3) ● (2) = 2 6 

This is easy to see if you expand the exponents: 

(2 3 ) 2 = 

(2³)(2³) = 

(2 ● 2 ● 2) (2 ● 2 ● 2) = 

2 ● 2 ● 2 ● 2 ● 2 ● 2 = 

2 6 

On Your Own 

(y³)² = ______ (2)² = ______ 

(n 3 ) 3 = _____ (5)³ = _____ 

(2²) = _____ (x²y)² = _____ 

66




Square Roots of Non-Perfect Squares 

Remember that when we multiply a whole number by itself, we get a 

perfect square. And the square root of a perfect square is the factor 

that, when multiplied by itself, gave us the perfect square. 

For example, the square root of 64 is 8 because 8 ● 8 = 64. 

But whole numbers that are not perfect squares still have square 

roots. However, their square roots are not whole numbers; rather 

they are decimals or fractions of whole numbers. 

On the CAHSEE, you may be given a non-perfect square and asked to 

place its root between two consecutive whole numbers. 

Example: Between what two consecutive whole numbers is 153 ? 

Solution: 

Think about our list of perfect squares. Refer to the chart on the next 

page. 

Since 153 falls between 144 and 169 in our perfect squares list, the 

square root of 153 is between 12 and 13. (Note: 12 and 13 are the 

square roots of 144 and 169 respectively). 

67




Memorize these for the CAHSEE! 

Number Square 

1 1 

2 4 

3 9 

4 16 

5 25 

6 36 

7 49 

8 64 

9 81 

10 100 

11 121 

12 144 

13 169 

14 196 

15 225 

16 256 

17 289 

18 324 

19 361 

20 400 

68

On Your Own 




1. Between what two consecutive whole numbers is the square root of 

17? 

2. Between which two consecutive whole numbers is 200 ? 

3. Between which two consecutive whole numbers is 130 ? 

4. The square root of 140 is between which two numbers? 

5. Between which two integers does 53 lie? 

69




Roots and Exponents 

On the CAHSEE, you may be given a variable that has been raised to a 

power and asked to find the base (the original number before it was 

raised). 

Example: If x² = 25, find the value for x. 

Since the base (x) is raised to the second power, we can find the value 

for x by taking the square root of x² . Since we have an equation, we 

must also find the square root of 25 so that the two sides of the 

equation remain in balance. 

x² = 25 ⎯ ⎯→ x = 5 

You may also be given the root of a variable and asked to find the 

variable. 

Example: x = 5 

To solve this, we need to square both sides of the equation: 

( x )² = (5)² ⎯ ⎯→ x = 25 

On Your Own: Find x: 

x² = 64 ______ x³ = 8 ______ 

x = 1/27 ______ x² = 9/16 ______ 

x = 8/27 ______ x² = 4/25 ______ 

x = 11 ______ 

3 x = 10 ______ 

x = 20 ______ x = 9 ______ 

70

Scientific Notation 




Scientific Notation is a way to express very small or very large 

numbers, using exponents. 

Example of a Very Big Number: The distance of the earth from the 

sun is approximately 144,000,000,000 meters. 

You can see that it can be tedious to write so many zeroes. This 

number can be expressed much more simply in scientific notation: 

1.44 X 10 

Example of a Very Small Number: An example using a very small 

number is the mass of a dust particle: 0.000000000 753 kg. 

We can write this number in scientific notation as 7.53 X 10 - º. 

On the CAHSEE, you will need to . . . 

• Read numbers in scientific notation 

• Compare numbers in scientific notation 

• Convert from standard notation (15,340) to scientific notation 

(1.534 X 10 4 ) 

• Convert from scientific notation (2.36 X 10¯ 3 ) to standard 

notation (.00236) 

71




Scientific Notation is a special type of exponent expression: the base 

is always 10 and it is raised to a positive or negative power. 

A number written in scientific notation consists of four parts: 

i. 4.95 X 10¯² 

a number (n) greater than or equal to 1 and less than 10 

ii. 4.95 X 10¯² 

iii. 4.95 X 10¯² 

iv. 4.95 X 10¯² 

a multiplication sign 

the base, which is always 10 

a positive or negative exponent 

72




Examples Correct Scientific Notation? Why? 

4.5 x 10 13 

45.6 x 10 -8 

Yes 

No 

1 ≤ n < 10 

n > 10 

Remember: For an expression to be written in correct scientific 

notation, the number (n) that appears before the base must be greater 

or equal to 1 and less than 10. 

On Your Own: Check all expressions in correct scientific notation: 

3.2 X 10 13 

23.6 X 10 12 

5.788 X 10³ 

5.788 X 10¯³ 57.88 X 10² 2.36 X 10³ 

2.36 X 10² 0. 0236 X 10 8 0. 236 X 10 7 

2.3 X 10 7 2. 3 X 10¯³ ` 0.23 X 10¯² 

73




Converting to Scientific Notation: 

Example: Write in scientific notation: 3,860,000 

(1) Convert to a number between 1 and 10. 

How: Place decimal point such that there is one non-zero digit to the 

left of the decimal point: 3.86 

(2) Multiply by a power of 10: 

How: Count number of decimal places that the decimal has "moved" 

from the original number. This will be the exponent of the 10. 

3 8 6 0 0 0 0 

6 5 4 3 2 1 

We have moved 6 places so the number (3.86) is multiplied by 10 6 

(3) If the original number was less than 1, the exponent is negative; if 

the original number was greater than 1, the exponent is positive. 

3,860,000 > 1, so the exponent is positive. 

Answer: 3.86 X 10 6 

74




On Your Own: Express in correct scientific notation: 

Standard Form Scientific Notation 

39,400 

0.0000394 

394 

39,400,000 

39.4 

0.394 

3,940 

0.00394 

3.94 

0.000394 

Place the following numbers in order, from smallest to largest: 

3.35 X 10 0 , 7.4 X 10 -2 , 1.6 X 10 -1 , 4.33 X 10 3 , 7.45 X 10 -3 

________ ________ ________ ________ _________ 

75




Converting from Scientific Notation to Standard Form: 

A. Positive Exponents 

• If exponent is positive, move decimal point to the right. 

• The exponent will determine how many decimals to move. 

Example: 3.45 X 10² = 345 (Move to the right 2 places) 

B. Negative Exponents 

• If exponent is negative, move decimal point to the left. 

• The exponent will determine how many decimals to move. 

Example: 3.45 X 10¯² = .0345 (Move to the left 2 places) 

On Your Own: Express in standard form: 

a. 3.45 x 10 -8 

• Since the exponent is negative, we move to the left. 

• Since the exponent is 8, we move to the left 8 places. 

Answer: ________________________________ 

b. 5.3 X 10³ ______________ 

c. 3. 5.3 X 10¯³ ______________ 

d. 7.98 X 10¯ 4 ________________ 

e. 7.98 X 10 5 ___________________ 

76

Unit Review 

1. (3)¯² = _____ 

2. (-3)² = _____ 

3. -(3)² = _____ 

3. (-3) 1 = _____ 

4. (3) 1 = _____ 

5. (3)¯ 1 = _____ 

6. (3) 0 = _____ 

7. (-3) 0 = _____ 

8. -(3) 0 = _____ 

9. 3 X 3 = _____ 




77

10. (3 1 )² = _____ 




11. 3 ● 10¯= _____ 

12. 3¯ ● 3= _____ 

13. 3 3 ● 3 -6 = _____ 

14. 

15. 

16. 

3 64 = _____ 

4 16 = _____ 

25 

= _____ 

100 

17. 4 4 X 4 0 = _____ 

−6 

3 

18. = _____ 

−8 

3 

19. 4 -4 X 4 4 = _____ 

78




20. Which shows the number 34,600,000 written in scientific 

notation? 

A. 346 X 10 5 

B. 34.6 X 10 6 

C. 3.46 X 10 7 

D. 3.46 X 10 -7 

E. 0.346 X 10 -8 

5 

3 

21. = _____ 

− 1 

3 

3 

22. 1 

3 

− 

5 

= _____ 

4 

4 

23. = _____ 

− 1 

4 

−5 

4 

25. = _____ 

−3 

4 

26. 5 3 ● 5 -2 = _____ 

79




Unit Quiz: The following problems appeared on the CAHSEE. 

3 ³ 

1. { } = _______ 

4 

9 

A. 

12 

9 

B. 

16 

27 

C. 

32 

27 

D. 

64 

1 

2. Solve for x: x³ = 

8 

A. x = 2 

B. x = 3 

1 

C. x = 

2 

1 

D. x = 

3 

3. Which number equals (2)¯ 4 

A. -8 

1 

B. − 

16 

1 

C. 

16 

1 

D. 

8 

80

4. 10 -2 = _____ 

10 -4 

A. 10¯ 6 

B. 10¯² 

C. 10² 

D. 10 




5. Between which two integers does 76 lie? 

A. 7 and 8 

B. 8 and 9 

C. 9 and 10 

D. 10 and 11 

6. The square of a whole number is between 1500 and 1600. The 

number must be between: 

A. 30 and 35 

B. 35 and 40 

C. 40 and 45 

D. 45 and 50 

7. The square root of 150 is between which two numbers? 

A. 10 and 11 

B. 11 and 12 

C. 12 and 13 

D. 13 and 14 

81




8. The radius of the earth’s orbit is 150,000,000,000 meters. What is 

this number in scientific notation? 

A. 1.5 X 10¯¹¹ 

B. 1.5 X 10¹¹ 

C. 15 X 10¹º 

D. 150 X 10 9 

9. 3.6 X 10² = ____ 

A. 3.600 

B. 36 

C. 360 

D. 3,600 

10. (3 8 ) 2 = _____ 

A. 3 

B. 3 6 

C. 3º 

D. 3 6 

11. 4³ X 4² = _____ 

A. 4 

B. 4 6 

C. 16 

D. 16 6 

12. (x 2 ) 4 = _____ 

A. x 6 

B. x 

C. x 6 

D. x² 

82




Unit 3: Multi-Step Word Problems 

Some problems involve more than one step. These are called multistep 

problems. On the CAHSEE you can expect to get at least a few 

multi-step problems. 

Example: The following problem appeared on the CAHSEE. 

The five members of a band are getting new outfits. Shirts cost $12 

each, pants cost $29 each, and boots cost $49 a pair. What is the 

total cost of the new outfits for all the band members? 

To solve this kind of problem, we must follow some basic steps: 

A. First, determine what the question asks: Total cost for all band 

members 

B. Write down all of the numerical information given in the problem: 

• 5 members in band 

• Shirts @ $12 each 

• Pants @ $29 each 

• Boots @ $49 each 

C. Determine the operations required to solve the problem. In other 

words, what do we do with all of the numbers listed in step 2? 

• Multiply each item bought by 5 since there are 5 members and 

each item is required for each member: 

Shirts: 12 X 5 = ________ 

Pants: 29 X 5 = ________ 

Boots: 49 X 5 = ________ 

• Add it all up (listing biggest numbers first): 

Answer: The total cost of the band’s outfits is _______. 

83

On Your Own 




1. Derrick wants to buy a sweater that costs $46. If he has $22 saved 

up and earns $12 a week in allowance, how long will it take before 

he has enough money to buy the sweater? 

Steps: 

A. What does the question ask: _______________________________ 

______________________________________________________ 

B. Write down all of the information that is important: 

• _______________________ 

• _______________________ 

• _______________________ 

C. Determine the operations required to solve the problem and then 

apply these operations to solve the problem. 

• The sweater costs $46, but he already has $22. How much more 

money does he need? Which operation is required to answer 

this question? _________________ 

Solve: 

• Now that we know how much more money Derrick needs, all we 

have to do is to figure out how many weeks it will take to earn 

this amount. Which operation is required to answer this 

question? _____________ 

Solve: 

Answer: Derrick can buy the sweater in ___ weeks. 

84




2. Uncle Bernie took his three nieces to the movies. Each niece 

ordered a small popcorn, a large soda, and a chocolate bar. If a 

small order of popcorn costs $4, a large soda costs $3, and a 

chocolate bar costs $1.50, how much did Uncle Bernie spend on 

snacks? 

3. Cynthia wants to buy a pair of jeans that cost $56, including tax. 

If she earns $10.50 each week for allowance and spends $3.50 

per week on bus fare to and from her dance lessons, what is the 

fewest number of weeks that it will take Cynthia to save enough 

money to buy the jeans? 

85




Extraneous Information 

Sometimes there may be information in the problem that you don’t 

need. It may be there to confuse you. Whenever you come to 

information that is extraneous (i.e., don’t need it, don’t want it), 

cross it out: 

Example: Daphne, Cynthia, and Rachel went to the movies on 

October 21. October 21 fell on a Friday. The movie began at 8:00 

p.m. They each bought a bucket of popcorn and a snickers bar. If 

each movie ticket costs $8.00, a bucket of popcorn costs $4.00, and a 

snickers bar costs $2.50, how much money did they spend altogether? 

Steps: 

• Cross out any information that you don’t need. Don’t just ignore it 

- - cross it out. 

Daphne, Cynthia, and Rachel went to the movies on October 21. 

October 21 fell on a Friday. The movie began at 8:00 p.m. They each 

bought a bucket of popcorn and a snickers bar. If each movie ticket 

costs $8.00, a bucket of popcorn costs $4.00, and a snickers bar costs 

$2.50, how much money did they spend altogether? 

• Write down all of the information that is important: 

3 people 

Tickets $8.00 each 

Popcorn $4.00 each 

Snickers $2.50 each 

• Figure out how much one person spent: 

8 + 4 + 2.50 = ______ 

• Figure out how much all three people spent: 

_____ ● 3 = _____ 

86

On Your Own 




1. Matthew bought a used car for $800. The car was 15 years old. He 

wrote a check for $520 and gave the salesman $125 in cash. The 

rest he promised to pay at the end of the week, when he would be 

receiving a paycheck for $385. How much does he owe on the car? 

• Cross out any information that you don’t need. Don’t just 

ignore it - - cross it out. That way you will be sure that you 

don’t accidentally slip it in later. 

• Now write down all of the information that is important: 

_____________________________________________ 

_____________________________________________ 

_____________________________________________ 

• How much did Matthew already pay? ____________ 

• How much more does he owe? _________ 

2. Mrs. Brown took her four children out to the pizza party. She and 

her children each ordered a small pepperoni pizza and a large soda. 

A small vegetarian pizza costs $4.50, while a small pepperoni pizza 

costs $5.25. Small sodas cost $2.00, medium sodas cost $2.50, 

and large sodas cost $3.25. How much did Mrs. Brown spend at 

the pizza party? 

87

Unit Quiz 




1. Martine bought 3 cans of soda for 65¢ each, 2 pretzels for $1.25 

each and a slice of pizza for $1.75. She paid with a $20 bill. How 

much change should she get back from the cashier? 

2. Joy spent 25% of her weekly paycheck to help her sister buy a new 

dress. If the dress costs $235 and her paycheck is $450, how 

much does she have left for the week? 

3. If it costs $150 to feed a family of four for the week, how much will 

it cost to feed a family of six? 

4. Adrienne has $166 left in her checking account and $1300 in her 

savings account. Each week she earns $175 as a cashier at the 

Five and Dime Store. She is planning on buying a set of dishes for 

her best friend’s wedding shower. The set costs $500. If she does 

not take any money out of her savings account, what is the fewest 

number of weeks that she must work in order to buy the dishes? 

88

Unit 4: Percent 




On the CAHSEE, you will have many problems that involve percent. 

(Note: Some of these will be word problems.) 

Percent of a Number 

Percent, written as %, literally means "out of 100." Any number 

expressed as a percent stands for a fraction. 

Example: Five percent (or 5%) means 5 out of 100. As a fraction, 

this is written as 5/100, which can be reduced to 1/20. This can 

also be expressed as a decimal: .05 (read as five hundredths). 

Example: Seventy-five percent (75%) means 75 out of 100, or 

75/100. This fraction can be reduced to 3/4. As a decimal, 75% 

would be written as 0.75, which means 75 hundredths. 

Converting from Percent to Decimal 

To change a percent to a decimal, divide the percent value by 100: 

move the decimal point two places to the left. 

Examples: 

17% = .17 

3% = .03 

80% = .80 (or .8) 

125% = 1.25 

.8% = .008 

3.4% = .034 

46% = 0.46 

89




Converting from Decimal to Percent 

To change a decimal to a percent, move the decimal point two places 

to the right. (Multiply by 100!) 

Examples: 

.34 = 34% .09 = 9% 2.3 = 230% .6 = 60% 0.125 = 12.5% 

Practice: Fill in the following chart. Reduce fractions to lowest terms. 

Fraction Decimal Percent 

9 

100 

9 

10 

1 

4 

0.08 

0.8 

0.84 

35% 

95% 

60% 

90




Solving Percent Problems 

There are two methods for solving percent problems. The first is 

setting up a proportion. 

Method 1: Proportion 

A proportion is two equivalent ratios, written as fractions. In 

any proportion, the product of the means is equal to the product 

of the extremes: 

We see that this is true: 3 ● 10 = 30 and 5 ● 6 = 30 

The product of the means is equal to the product of the extreme. 

We can solve a percent problem by setting up a proportion. Here is 

the proportion used to solve percent problems: 

Part x 

= 

Whole 100 

This proportion may be translated as follows: The part is to the whole 

as what number is to 100?" 

And, since in a proportion, the product of the means is equal to the 

product of the extremes, the following is also true: 

Whole ● x = Part ● 100 

This relationship will always be true for a proportion. Since we 

multiply diagonally across the proportion, people often use the term 

"cross multiplying" for short (since it can be encumbering to keep 

saying, "The product of the means is equal to the product of the 

extremes." You can use the term "cross-multiplication" if you like; 

just remember the concept that is behind it. 

91




Let's solve a percent problem together, using the proportion method: 

Example: 23 is what percent of 50? 

23 x 

= 

50 100 

50x = 2300 ← ⎯⎯ Now divide to solve for x. 

x = ___ 

23 

Answer: is equal to ___% 

50 

On Your Own: Use the method of cross multiplication to solve for x. 

x 5 

1. = 

20 100 

92

4 x 

2. = 

25 100 

18 x 

3. = 

20 100 

7 x 

4. = 

10 100 




93




Let's look at more examples of percent problems solved with the 

proportion method: 


• Set up a proportion: 

Part x 

= 

Whole 100 

Note: The term 50 is the "whole" and 18 is the "part": 

18 x 

Proportion: = 

50 100 

• Cross multiply: 50x = 1800 

1800 

• Isolate x by dividing by 50: x = = ____ 

50 

Example: What is 25% of $60? 

x 25 

• Set up a proportion: = 

60 100 

Note: The term 60 is the "whole" and the problem asks for 25% 

of this whole: 

• Cross multiply: 100x = 1500 

1500 

• Find x: x = = ----- 

100 

94

On Your Own 




1. 25% of what number is 15? 

15 25 

• Set up a proportion: = 

x 100 

Note: The term 15 is the "part" and the problem asks for 25% of 

the whole: 

• Cross multiply: __________________ 

• Find x: _______________________ 

2. 30 is what percent of 50? 

3. What is 45% of 90? 


95




Method 2: Translate & Compute 

In this method, we translate the problem into an algebraic equation 

and then solve. Let's look at the previous problems again, this time 

using this second method: 


• Translate: 

18 is what percent of 50? 

x 

18 = ● 50 

100 

50x 

• Compute: 18 = 

100 

18 = 1 50x ← ⎯⎯ Simplify all fractions 

2100 

x 

18 = 

2 

18 ● 2 = x ● 2 1 ← ⎯⎯ Multiply both sides by 2. 

12 

x = ___ Answer: 18 is __% of 50. 

Note: This method can also be used as a complement to Method 1 

and as way to verify the answers you arrived at using Method 1. 

96

On Your Own 




1. What is 25% of $60? 


What is 25 percent of 60? 

___ ___ ____ ___ ___ 

• Compute: 



25 percent of what number is 15? 

____ ___ ___ ___ ___ 

• Compute: 

97






5. What is 15% of 90? 


98




Practice: Use either method to solve the percent problems below. 

1. On average, every 16 out of 200 students study Calculus. What 

percent study Calculus? 

2. If 150 people were surveyed in the 2004 presidential elections and 

90 of those people said that they were going to vote for John Kerry, 

find the percent of Kerry supporters in the sample population? 

3. The Brighton Movie Theater sells the following candies at their 

snack bar: Snickers Candy Bars, Peanut Clusters, O’Henry Bars, 

M & M’s, and Milky Way Bars. Currently, there are 10,000 candies 

in stock. The following chart below shows a percentage breakdown 

of each type of candy in stock. Find the actual number of each type 

of candy: 

Candy Number 

Snickers: 5% 

Peanut Clusters: 15% 

O’Henry: 20% 

M & M’s: 25% 

Milky Way Bars: 35% 

4. What is the fractional equivalent of 95% (reduced to lowest terms)? 

99




5. Four months of the year have 30 days. Which percentage most 

closely represents the months that do not have 30 days? 

A. 33% 

B. 44% 

C. 66% 

D. 75% 

12 

6. What is the percentage equivalent of ? 

15 

7. In Mr. Martin’s class, 9 of the 27 students in Mr. Martin’s class 

received a B+ or higher on the Algebra quiz. What percent of the 

students received a grade of B or lower? 

4 

8. What is expressed as a percent? 

5 

9. What is 0.80 expressed as a fraction (in simplest terms)? 

100




Percent Increases & Decrease 

Often, when we compare one entity across time (such as changes 

in population or the price of a particular item), we express these 

changes in terms of percent. The percent of change is the ratio of 

the amount of change to the original amount. 

Ratio: How much it went up or down 

The original amount 

An easy way to solve percent increase & percent decrease problems is 

to set up a proportion that consists of two ratios, the one above and 

a second one for the percent. Remember that “percent” is always a 

ratio and the denominator of that ratio is always 100. 

What 

So the second ratio looks like this: 

Percent 

x 

or 

100 

If you set up a proportion using these two ratios, you get the 

following: 

How much it went up or down = x_ 

The original amount 100 

All you need to do after that is cross multiply and isolate the x value. 

Let's look at an example on the next page. 

101




Example: Jimmy got a raise from $6.00 to $8.00 per hour. This 

represents a raise of what percent? 

Steps: 

• Find out how much it went up or down and place this number 

over the original amount: 

2 ← ⎯⎯ It went up $2 

6 ← ⎯⎯ Original amount is $6.00 

• Set up proportion: 

2 x 

= 

6 100 

• Cross multiply: 

• Solve for x: 

• Express answer as a percent: _________ 

Make sure you use the original amount as the denominator! 

102

Alternative Method 




You can also solve the previous problem by setting up an equation: 

Change = What Percent of Original Amount 

Let's look at the problem again: 

Example: Jimmy got a raise from $6.00 to $8.00 per hour. This 

represents a raise of what percent? 

Now plug the values in the equation: 


x 

2 = ● 6 

100 

Now solve: 

x 

2 = (6) 

100 

103

On Your Own 




1. Tasty Delight raised their prices on ice cream sundaes from $5.00 

to $7.00. This represents an increase of what percent? 

x 

• Set up proportion: ____= 

100 

• Cross multiply: _________________ 

• Solve for x: _____________________ 

• Express answer as a percent: ________ 

Note: Be sure to use the original amount as the denominator! 

Now solve the above problem using the alternative method: 

Plug in values: 


____ ____ ___ ______ 

Solve: 

104




2. A shirt that costs $40 in 1999 costs $60 in 2005. What is the 

percent increase? (Use either method to solve.) 

Solve: 

Percent Increase: _______________ 

What would be a trick answer on the CAHSEE? _____________ 

3. Elizabeth’s basketball card collection increased in value from $500 

to $1,000. What is the percent increase? (Use either method to 

solve.) 

Solve: 

Percent Increase: _______________ 

What would be a trick answer on the CAHSEE? ____________ 

105




4. Jordy’s basketball card collection decreased in value from $1,000 to 

$500. What is the percent decrease? 

Solve: 

Percent Decrease: ____________________ 

What would be a trick answer on the CAHSEE? _______________ 

5. Last year Andrea had 36 students in her class. This year she only 

has 27. What is the percent decrease? 

Solve: 

Percent Decrease: ______________________ 

What would be a trick answer on the CAHSEE? _______________ 

6. If a shirt that costs $80 last year is worth only 75% as much this 

year, what is the current value of the shirt? 

Solve: 

Answer: _________ 

106

Price Discounts 




Stores will often sell items for a discounted sales price. The store will 

discount an item by a percent of the original price. To find the amount 

of a discount (in dollars), simply multiply the original price by the 

percent discount. 

Example: An item, which originally cost $20, was discounted by 25%. 

Find the discount in dollars? 

Steps: 

• Translate the problem into math: 

25 

25% of $20 = ● 20 

100 

• Calculate: 

25 

● 20 = $5.00 (or ¼ ● 20 = 5) 

100 

The item was sold for $5.00 less than its original price. 

Terms you may see for discounted items: 

• 50% Off 

• Save 50% 

• Discounted by 50% 

107

On Your Own 




1. A transistor radio is normally sold for $80 at Karter Electric Goods. 

This week, it is being offered at a 20% discount. How much 

cheaper is the radio this week? 

2. A dress, which sold for $80 last week, is on sale for 20% off. This 

represents a discount of how much (in dollars)? 

3. At Peppy’s Pizza, a small pepperoni pizza normally sells for $6.00. 

This week, the store is offering a 25% discount on all small pizzas. 

How much cheaper is the pizza this week? 

108




Note: Many price discount questions on the CAHSEE ask you to find 

the new price, not the amount of discount. This involves one 

more important step. 

Example: An item originally cost $20 and was discounted by 25%. 

What was the new sales price? 

Steps: 

25 

• Translate the problem into math: 25% of $20 = ● 20 

100 

• Calculate the discount in dollars: 

25 

● 20 = $5.00 (or ¼ ● 20 = 5) 

100 

The item was sold for $5.00 less than its original price. 

• Finally, to find the new sales price, subtract the amount of discount 

from the original price: 

$20.00-$5.00=$15.00 

CAHSEE Alert! Don’t forget this last step. If this were an actual item 

on the exam, $5.00 (the amount of discount) would probably be one of 

the answer choices. If you are working out a problem that has multiple 

steps, remember to do all of the steps to get the right answer. 

109

On Your Own 




1. A dress, which sold for $80 last week, is on sale for 20% off. What 

is the new price of the dress? 

Solve: 

New Price: ________ 

What would be a trick answer on the CAHSEE? ____________ 

2. At Peppy’s Pizza, a small pepperoni pizza normally sells for $6.00. 

This week, the store is offering a 25% discount on all small pizzas. 

How much does a small pepperoni pizza cost this week? 

Solve: 


What would be a trick answer on the CAHSEE? ___________ 

3. Rain boots regularly sell for $70 a pair. They are currently on sale 

for 40% off. What is the sale price of the boots? 

Solve: 


What would be a trick answer on the CAHSEE? ___________ 

110

Markups 




Stores buy items from a wholesaler or a distributor and increase the 

price when they sell them to consumers. The increase in price provides 

money for the operation of the store and the salaries of people who 

work in the store. A store may have a rule that the price of a certain 

type of item needs to be increased by a certain percentage. This 

percentage is called the markup. 

A. Two-Step Method 

1. Find the markup in dollar amount: 

Original Cost ● Percent of Markup 

2. Add this dollar amount to the original price. 

Markup in $ + Original Price 

Example: A merchant buys an item for $4.00 and marks it up by 

25%. How much does he charge for the item? 

1. Find the markup in dollars: 

$4.00 ● .25 = $1.00 Or . . . 

1 

4 ● = 1 

4 

2. Add this dollar amount to the original price: 

$4.00 + 1.00 = $5.00 

CAHSEE Alert! Like the discount problems, be careful not to forget 

the last step. In the above problem, a probable answer choice would 

be $1.00. Don’t be fooled! Read the question carefully. 

111




B. One-Step Method 

A faster way to calculate the sales price is to make the original cost 

equal to 100%. 

Example: A merchant buys an item for $4.00 and marks it up by 

25%. How much does he charge for the item? 

Since the markup is 25%, the customer pays 125% of the original 

cost. Multiply the original cost by 125% (or 1.25): 

$4.00 ● 1.25 = 

(4 ● 1) + (4 ● .25) = 

4 + 1 = $5.00 The merchant charges $5.00 for the item. 

CAHSEE Alert! Some questions on the CAHSEE may ask for the 

dollar amount of the markup, not the final sales price. 

Example: Harry’s Bargain Basement has a 20% markup on all its 

goods. If the manufacturer price of irons is $16, how much extra 

does the customer pay for each iron? 

Solve: _____________________________________________ 

Compare with this problem: 

Harry’s Bargain Basement has a 20% markup on all its goods. If the 

manufacturer price of irons is $16, how much does the customer pay 

for an iron? 

Solve: _____________________________________________ 

Note: On the CAHSEE, be sure to read the question carefully to 

determine whether it is asking for the markup or final sales price. 

112




On Your Own: Read each problem carefully and determine what the 

question is asking. Then solve the problem. 

1. The original cost of a dress is $12.00. (This is the amount that the 

store paid the manufacturer.) The store marks up all their items by 

20%. How much does the store charge for the dress? 

2. Bill’s Auto Supplies buys tires for $80. If the store sells its tires for 

$100, what is its percent markup? 

3. All items at Bargain Slim’s have been marked up by 40%. If the 

store paid $12 for each CD, how much does the customer pay? 

4. A stainless steel refrigerator is bought for $500 and then marked up 

by 100%. What is the new price of the refrigerator? 

113

Commissions 




Sales commissions are often paid to employees who sell merchandise 

or products. Commissions serve to motivate salespersons to sell a lot. 

A commission is generally a percentage of the total sales made by 

a salesperson. To find the commission, just multiply the value of 

the total sales by the commission rate. It is that simple! 

Example: A salesman receives a 10% commission on all sales. If he 

sells $1500 worth of merchandise, how much does he earn in 

commission? 

$1,500 X 0.10 = $150 

or 

$1500 X 10_ = 15 X 10 = $150 

100 

On Your Own 

1. Sarah is a real estate agent. She earns 12% commission on every 

house she sells. Sarah recently sold a house for $400,000. What 

was her commission? 

2. Ronald is a salesman in the men’s department at Bloomingdale’s 

Department Store. He earns 15% on all sales. His total sales for 

the month of August came to $80,000. How much did he earn in 

commission? 

7. Alvin Ray sells used cars at Kaplan’s Auto Dealer. His commission 

rate is 30%. What was his commission on the used Audi he sold for 

$36,000? 

114




Unit Quiz: The following questions appeared on the CAHSEE. 

1. At a recent school play, 504 of the 840 seats were filled. What 

percent of the seats were empty? 

A. 33.6% 

B. 40% 

C. 50.4% 

D. 60% 

2. Some of the students attend school 180 of the 365 days in a year. 

About what part of the year do they attend school? 

A. 18% 

B. 50% 

C. 75% 

D. 180% 

3. What is the fractional equivalent of 60%? 

1 

A. 

6 

3 

B. 

6 

3 

C. 

5 

2 

D. 

3 

115




4. If Freya makes 4 of her 5 free throws in a basketball game, what is 

her free throw shooting percentage? 

A. 20% 

B. 40% 

C. 80% 

D. 90% 

5. Between 6:00 AM and noon, the temperature went from 45° to 90°. 

By what percentage did the temperature increase between 6:00 AM 

to noon? 

A. 45% 

B. 50% 

C. 55% 

D. 100% 

6. The price of a calculator has decreased from $12.00 to $9.00. 

What is the percent of decrease? 

A. 3% 

B. 25% 

C. 33% 

D. 75% 

7. The cost of an afternoon movie ticket last year was $4.00. This 

year an afternoon movie ticket costs $5.00. What is the percent 

increase of the ticket from last year to this year? 

A. 10% 

B. 20% 

C. 25% 

D. 40% 

116




8. A pair of jeans regularly sells for $24.00. They are on sale for 25% 

off. What is the sales price of the jeans? 

A. $6.00 

B. $18.00 

C. $20.00 

D. $30.00 

9. A CD player regularly sells for $80. It is on sale for 20% off. What 

is the sales price of the CD player? 

A. $16 

B. $60 

C. $64 

D. $96 

10. Mr. Norris is paid a 5% commission on each house that he sells. 

What is his commission on a house that he sells for $125,000? 

A. $625 

B. $6,250 

C. $62,500 

D. $625,000 

117

Unit 5: Interest 




On the CAHSEE, you may be asked several questions on interest. 

These questions will cover both simple interest and compound interest. 

Introduction to Interest 

Did you know that money can make more money? Whenever money is 

invested or borrowed, additional funds, called interest, are 

charged for the use of that money for a certain period of time. When 

the money is paid back, both the principal (amount of money that 

was borrowed) and the interest are due. 

If you invest money in the bank, the bank is borrowing the money and 

the interest is paid to you. On the other hand, when you take out a 

loan, you are borrowing the money and you must pay the interest. 

Interest can be simple or compound: 

Simple interest is generally used when borrowing or investing money 

for short periods of time. 

Compound interest is generally used when borrowing or investing 

money for longer periods of time. We will learn about compound 

interest later. 

Interest depends on three things: 

1. Principle (P): The amount you invest or borrow; principle is 

expressed in dollars. 

2. Interest Rate (R): How much it costs you to borrow the money or 

how much you gain by investing your money; this rate is always 

expressed as a percent in the problem, although you may convert 

the rate to a decimal during computation. 

3. Time (T): How long you borrow the money or how long you invest 

your money; this time is always expressed in number of years 

118




Converting to Years 

When calculating interest, time is expressed in years. If the period 

of time is given in months, you must first convert it to the number of 

years. 

Examples: 

1 

6 months is year, or .5 year. 

2 

3 

18 months is years, or 1.5 years. 

2 

Working with Improper Fractions 

We will see that, when calculating interest, it is easiest to work with 

improper fractions than with mixed numbers. 

Mixed Numbers: A mixed number consists of both a whole integer 

and a fraction. 

1 

1 is a mixed number because it consists of a whole number (1) and 

4 

1 

a fraction . 

4 

Improper Fractions: An improper fraction is one in which the 

numerator is greater than the denominator. 

4 

5 is an improper fraction because the numerator (5) is greater than 

the denominator (4). 

119




Converting Mixed Numbers to Improper Fractions 

To convert a mixed number to an improper fraction, follow these 

steps: 

• Multiply the whole number by the denominator of the fraction. 

• Add the numerator of the fraction to the product found above. 

• Place the result over the fraction's denominator. 

1 

Example: Convert 1 to an improper fraction. 

4 

• Multiply the whole number by the denominator of the fraction: 

1 ● 4 = 4 

• Add the numerator of the fraction to the product found in Step 1: 

4 + 1 = 5 

• Place the result over the fraction's denominator: 

5 

4 

3 

On Your Own: Convert 1 to a mixed fraction. 

5 

120




Practice: Express each time interval below in years. Express as both 

a fraction (reduced to lowest terms) and a decimal. 

Note: Be sure to covert any mixed number to an improper fraction. 

Months Years: Fraction Years: Decimal 

3 months 

4 months 

8 months 

27 months 

15 months 

9 months 

21 months 

1 year and 8 

months 

121




Converting to Percents & Decimals 

The interest rate is expressed as a percent or as a decimal. 

Examples: 

5 

5% = = 0.05 

100 

8. 

5 

8½% = 

100 

= 0.085 

x 

On Your Own: Express as both a percent ( ) and a decimal. 

100 

Rate Percent Decimal 

12% 

20% 

3% 

18% 

9½% 

12½% 

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Solving Simple Interest Problems 

To solve simple interest problems, just apply the formula: 

Principle ● Rate ● Time 

We can abbreviate this as follows: P ● R ● T 

Note: Be sure to learn this formula for the CAHSEE! 

One Step-Problems 

One-step problems ask you to find the interest (in dollars and cents) 

earned (from an investment) or owed (on a loan). Just apply the 

formula for interest: 

One-step problems ask you to find the interest (in dollars and cents) 

earned (from an investment) or owed (on a loan). Just apply the 

formula for interest: 


Note: Be sure to convert all terms to their correct units: 

• Rate in % or decimal 

• Time in years 

Example: $500 invested for 6 months in an account paying 7% 

interest. How much is earned in interest? 

To solve, simply plug the correct values into the equation and do the 

computation: 


$500 ● 0.07 ● 0.5 = _______ OR 

500 ● 7_ ● 1 = ______ 

100 2 

123




Fractions and Interest Problems 

As we mentioned earlier, when solving interest problems, it is easier to 

work with improper fractions than with mixed numbers. 

Example: Shawn invests $4,000 at 16%. How much does he earn in 

15 months? 

Let's solve this problem by expressing 15 months as an improper 

fraction. (Note: It is much easier to multiply with improper fractions 

than with mixed numbers.) 

There are 12 months in 1 year; we have 15 months: 

15 

← ⎯⎯ Improper fraction: Denominator > numerator 

12 

15 5 

We can reduce this fraction: = 

12 4 

Now let's solve the problem: 

P ● R ● T 

4000 ● 16 ● 5 = 

100 4 

4,000 ● 4 16 ● 5 = 

100 41 

40 ● 20 = ____ 

124




On Your Own: Plug the correct values and solve: 

1. $1,500 is borrowed at an interest rate of 3 percent for 20 years. 

How much is earned in interest? 

P = __________ 

R = __________ 

T = __________ 

P ● R ●T = _______________________________ 

2. $5,000 is invested for 24 months in an account paying 6% interest. 

How much is earned in interest. 

P = __________ 

R = __________ 

T = __________ 

P ● R ●T = _______________________________ 

3. Drew earns 6% in simple interest. If he invests $8,000 in a bank 

account, how much interest will he have earned after 18 months? 

Note: In each of the above questions, we are asked to find the 

interest earned, rather than the value of the entire investment. We will 

now learn to add the interest to the principle. 

125




Two-Step Problems 

On the CAHSEE, you may be asked to find the value of the entire 

investment. For these problems, there is one additional step. 

Example: Marianne invested $5,000 in the bank at an annual interest 

rate of 8 ½ percent. How much will her investment be worth in two 

years? 

• Find the amount of interest earned: 

P ● R ● T 

8. 

5 

5,000 ● ● 2 = _______________________ 

100 

• Add the interest to the principle to get the value of the investment: 

5,000 + _______ = $ ________ 

On Your Own: 

1. Rachel invests $3,000 at 12%. How much will her investment be 

worth in 15 months? 

• Plug the values into the formula and compute interest earned: 

• Add interest to principal: 

3000 + ___ = ____ 

126




2. Amy has a bank account that pays an annual interest rate of 7% 

(simple interest). If she has $7,000 in principal, how much interest 

will she earn this year? 

3. Denise earns 8% in simple interest each year. If she now has $900 

in her savings account, what will be the value of her savings 

account in six months? 

4. Emily has invested $15,000 at Chase Manhattan Bank. If her 

current rate of interest is 8%, how much interest will she have 

earned in nine months? 

5. Refer back to the previous problem. What will be the total value of 

Emily’s investment after nine months? 

127




Compound Interest 

While simple interest is paid once per year, compound interest can be 

paid twice a year (semi-annually), four times a year (quarterly) or 

even monthly! 

Example: Peter invests $500 in a savings account. The bank pays 

10% annual interest, compounded twice a year. What is the value of 

Peter’s investment after one year? 

Steps: 

• How much does Peter earn after 6 months? Not 10% because that 

is what he earns annually. 10% annual interest compounded twice 

a year means that half of the interest is paid after 6 months (half of 

the year) and the other half is paid at the end of the year. Since six 

months is one-half of a year, Peter only earns half of 10%, or 5%, 

after six months. 

Calculate 5% of $500: _5 ● 500 = $25. 

100 

• Add this to the principle to find the total value of his investment 

after six months: 

$500 + $25 = $525 

• For the next six months, Peter will earn 5% on $525. Calculate the 

interest: 

5% ● ____ = ______ 

• Add this amount to the value of his investment after one year: 

______ + _____ = ________ 

128




CAHSEE Tip: Since the math section of the CAHSEE uses a multiple- 

choice format, you can automatically rule out certain choices on 

compound interest problems: 

• We know that compound interest is always greater than simple 

interest; therefore, you can cross out any answers that are less 

than or equal to the amount calculated for simple interest. 

• At the same time, the answer will be only slightly greater than 

the amount obtained under simple interest since CAHSEE compound 

interest problems will generally be limited to one year (the 

difference between the interest earned under simple and compound 

interest gets bigger with each year); therefore, you can cross out 

answers that are significantly greater than that obtained under 

simple interest. 

See if you can apply this strategy for the following two problems. 

1. Ellie invested $3,000 in a savings account that pays an annual 

interest rate of 6% compounded twice a year. How much will she 

have in the bank after one year? 

A. $3,000.00 

B. $3,180.00 

C. $3,182.70 

D. $3,600 

2. Drew has invested $10,000 at Bank of America. His current rate of 

interest is 5%, compounded twice a year. How much interest will he 

have earned in one year? 

A. $10, 756.25 

B. $10,506.25 

C. $10,256.25 

D. $10,500 

129




Unit Quiz: The following questions appeared on the CAHSEE. 

1. Sally puts $200 in a bank account. Each year the account earns 

8% simple interest. How much interest will she earn in three 

years? 

A. $16.00 

B. $24.00 

C. $48.00 

D. $160.00 

2. Mr. Yee invested $2000 in a savings account that pays an annual 

interest rate of 4% compounded twice a year. If Mr. Yee does not 

deposit or withdraw any money, how much will he have in the bank 

after one year? 

A. $2,080.00 

B. $2,080.80 

C. $2,160.00 

D. $2,163.20 

Note: See if you can solve this problem by applying the multiplechoice 

strategy for compound interest problems. 

130

Number Sense

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